A refined Lusin-type gradient theorem for arbitrary Radon measures is proved, with an L^p estimate independent of the exceptional set when the datum is orthogonal to the decomposability bundle, implying the 1-dimensional flat chain conjecture and reducing the general conjecture to a statement…
Non-rigidity of the absolutely continuous part of $\mathcal{A}$-free measures
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abstract
We generalize a result by Alberti, showing that, if a first-order linear differential operator $\mathcal{A}$ belongs to a certain class, then any $L^1$ function is the absolutely continuous part of a measure $\mu$ satisfying $\mathcal{A}\mu=0$. When $\mathcal{A}$ is scalar valued, we provide a necessary and sufficient condition for the above property to hold true and we prove dimensional estimates on the singular part of $\mu$. Finally, we show that operators in the above class satisfy a Lusin-type property.
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2024 1verdicts
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A refined Lusin type theorem for gradients
A refined Lusin-type gradient theorem for arbitrary Radon measures is proved, with an L^p estimate independent of the exceptional set when the datum is orthogonal to the decomposability bundle, implying the 1-dimensional flat chain conjecture and reducing the general conjecture to a statement…